New solutions for nonlinear Schrödinger equations with critical nonlinearity

Abstract

We consider the following nonlinear Schrödinger equations in R n { ε 2 Δ u − V ( r ) u + u p = 0 in R n ; u > 0 in R n and u ∈ H 1 ( R n ) , where V ( r ) is a radially symmetric positive function. In [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, Part I, Comm. Math. Phys. 235 (2003) 427–466], Ambrosetti, Malchiodi and Ni proved that if M ( r ) = r n − 1 ( V ( r ) ) p + 1 p − 1 − 1 2 has a nondegenerate critical point r 0 ≠ 0 , then a layered solution concentrating near r 0 exists. In this paper, we show that if p = n + 2 n − 2 and the dimension n = 3 , 4 or 5, another new type of solution exists: this solution has a layer near r 0 and a bubble at the origin.